Homeostasis in gene regulatory networks

Gene regulatory networks lie at the heart of many important intracellular signal transduction processes. A Gene Regulatory Network (GRN) is abstractly defined as a directed graph, where the nodes represent genes and the edges represent the causal regulatory interactions between genes. It can be used to construct mathematical models describing the time-varying concentrations of several molecular species attached to each gene (node) in the network. In the deterministic setting, this is typically implemented by a system of ordinary differential equations. A biological system exhibits homeostasis when there is a target quantity, called the input–output function, whose values stay within a narrow range, under relatively wide variation of an external parameter. A strong form of homeostasis, called infinitesimal homeostasis, occurs when the input–output function has a critical point. In this paper, we use the framework of infinitesimal homeostasis to study general design principles for the occurrence of homeostasis in gene regulatory networks. We assume that the dynamics of the genes explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. Given a GRN, we construct an associated Protein-mRNA Network (PRN), where each individual (mRNA and protein) concentration corresponds to a node and the edges are defined in such a way that the PRN becomes a bipartite directed graph. By simultaneously working with the GRN and the PRN, we are able to apply our previous results about the classification of homeostasis types (i.e. topologically defined homeostasis generating mechanism) and their corresponding homeostasis patterns. Given an arbitrarily large and complex GRN [Formula: see text] and its associated PRN [Formula: see text], we obtain a correspondence between all the homeostasis types (and homeostasis patterns) of [Formula: see text] and a subset of the homeostasis types (and homeostasis patterns) of [Formula: see text]. Moreover, we completely characterize the homeostasis types of the PRN that do not have GRN counterparts.